---
title: Inexact Quantum Linear Programming Solver
url: https://www.emergentmind.com/topics/inexact-quantum-linear-programming-solver
type: topic
---

# Inexact Quantum Linear Programming Solver

An inexact quantum linear programming solver is a hybrid algorithmic framework for linear optimization in which the Newton systems arising from a barrier or interior-point method are solved by a Quantum Linear System Algorithm (QLSA), while the resulting search direction is treated as approximate because quantum state preparation, finite-precision inversion, and tomography introduce error. In "A quantum dual logarithmic barrier method for linear optimization," this framework is instantiated as an inexact-feasible quantum variant of the dual logarithmic barrier method (DLBM): a QLSA is used at each barrier iteration, the algorithm converges quadratically toward the central path with inexact directions, and it retains the best-known $\mathcal{O}(\sqrt{n}\log(n\mu_0/\zeta))$ iteration complexity for the stated setting [2412.15977].

## 1. Standard linear program and dual barrier geometry

The underlying optimization problem is the standard primal-dual pair
\[
(\mathrm{P})\quad \min_x c^T x \;\;\text{s.t.}\;\; Ax=b,\;x\ge 0,
\]
\[
(\mathrm{D})\quad \max_{y,s} b^T y \;\;\text{s.t.}\;\; A^T y+s=c,\;s\ge 0.
\]

For a barrier parameter $\mu>0$, the dual logarithmic barrier is
\[
\mathcal{L}(y;\mu)=-b^T y-\mu\sum_{i=1}^n \log\!\bigl(c_i-(A^T y)_i\bigr).
\]
Writing $S=\mathrm{Diag}(s)=\mathrm{Diag}(c-A^T y)$, its gradient and Hessian are
\[
\nabla \mathcal{L}(y)=-b+\mu\,A\,(c-A^T y)^{-1},
\qquad
\nabla^2 \mathcal{L}(y)=\mu\,A\,S^{-2}A^T.
\]

This formulation places the linear-algebraic core of the method in the symmetric positive-definite matrix $A S^{-2} A^T$. In the DLBM, the iterates are organized around the dual central path through the barrier parameter $\mu$ and the slack matrix $S$; consequently, the quality of each Newton solve directly determines whether the method preserves positivity and retains fast local convergence [2412.15977].

## 2. Exact dual logarithmic barrier method

At a dual iterate $(y,s=c-A^T y)$, the residual is
\[
r_p=b-\mu\,A\,S^{-1}e.
\]
The exact Newton system for $\Delta y$ is the normal-equation form
\[
(AS^{-2}A^T)\,\Delta y=\frac{1}{\mu}\,r_p.
\]
Once $\Delta y$ is obtained, the slack correction is
\[
\Delta s=-A^T\Delta y.
\]

With the full Newton update
\[
y^+=y+\Delta y,\qquad s^+=s+\Delta s,\qquad \mu^+=(1-\theta)\mu,
\]
the local progress measure
\[
\delta(s,\mu)=\|s^{-1}\Delta s\|_2
\]
satisfies
\[
\delta(s^+,\mu^+)\le \delta(s,\mu)^2
\]
provided that $\delta(s,\mu)\le 1$. The exact DLBM therefore exhibits quadratic contraction of the proximity measure. One shows that $\mathcal{O}(\sqrt{n}\log(n\mu^0/\epsilon))$ iterations suffice for gap at most $\epsilon$ [2412.15977].

This exact scheme is the benchmark against which the inexact quantum version is measured. The central question is whether the quadratic local behavior and short-step iteration bound survive once $\Delta y$ is no longer computed exactly.

## 3. Inexact-feasible variant and preservation of positivity

In the quantum setting, the Newton direction is only available approximately. The DLBM is therefore reformulated as an inexact-feasible method by writing
\[
\Delta \hat y=\Delta y+E_{\Delta y}^C,
\qquad
\Delta \hat s=-A^T\Delta \hat y=\Delta s+E_{\Delta s}^C.
\]

The key feasibility result is that positivity and quadratic local convergence persist under explicit inexactness bounds. In particular, Theorem 3.1 states that if
\[
\delta(s,\mu)\le \frac12
\qquad\text{and}\qquad
\|s^{-1}E_{\Delta s}^C\|_2\le \frac13\,\delta(s,\mu)^2,
\]
then
\[
s^+=s+\Delta s+E_{\Delta s}^C\ge 0
\]
and
\[
\delta(s^+,\mu)\le 1.5\,\delta(s,\mu)^2.
\]

For global complexity, the algorithm adopts
\[
\theta=\frac{1}{4\sqrt{n}}.
\]
Then Theorem 3.3 shows that the inexact-feasible DLBM still terminates in
\[
K\le \left\lceil 4\sqrt{n}\log(n\mu^0/\epsilon)\right\rceil
\]
full Newton-step iterations, achieving a duality gap
\[
x^T s\le 2\epsilon
\]
[2412.15977].

This establishes the defining feature of an inexact quantum LP solver in the DLBM setting: the approximation error is not treated as an implementation nuisance but as a first-class analytical object. A plausible implication is that the viability of the quantum acceleration hinges less on abstract QLSA asymptotics alone than on the compatibility between approximation error, feasibility, and barrier-path contraction.

## 4. Quantum linear-system subroutine and tomography-induced error

Each barrier iteration requires solving
\[
M=AS^{-2}A^T,\qquad M\,\Delta y=\frac{1}{\mu}r_p.
\]
The quantum subroutine proceeds by block-encoding $AS^{-1}$ in QRAM as a unitary $U$ and applying QSVT, in the Childs-Kothari-Somma style, to solve the Hermitian system
\[
(AS^{-1})(AS^{-1})^T\,\Delta y=\frac{1}{\mu}\,S^{-1}A\,r_p.
\]

The quantum solver outputs a normalized state $|\bar{\Delta y}\rangle$ with error $E_{\Delta y}^Q$ in $2$-norm. This state is then rescaled through
\[
\lambda^*=\frac{r_p^T\bar{\Delta y}}{\mu\,\|S^{-1}A^T\bar{\Delta y}\|_2^2}
\]
to form the approximate classical direction
\[
\Delta \hat y=\lambda^* \bar{\Delta y}.
\]
A final tomography step converts $|\bar{\Delta y}\rangle$ into a classical vector within error $\epsilon_{\Delta y}^Q$.

Tomography is analytically nontrivial because it feeds directly into the slack error. To maintain feasibility, the paper imposes
\[
\|\epsilon_{\Delta y}^Q\|_2\le \frac{0.005/1.995}{\sqrt{\kappa(AS^{-2}A^T)}},
\]
which guarantees
\[
\|s^{-1}E_{\Delta s}^C\|_2\le 0.1\,\delta(s,\mu).
\]
Theorem 4.6 gives the QRAM-query complexity of each QLSA-plus-tomography call as
\[
\widetilde{O}_{n,\|AS^{-1}\|_F,\kappa(AS^{-1}),1/\epsilon_{\Delta y}^Q}
\!\left(\frac{m\,\kappa^2(AS^{-1})}{\epsilon_{\Delta y}^Q}\right),
\]
with classical overhead $O(mn)$ per iteration [2412.15977].

The dependence on tomography error is central. This suggests that the solver is intrinsically hybrid: the quantum component prepares an approximate direction efficiently in amplitude form, but the classical optimization loop ultimately depends on extracting that direction with enough fidelity to satisfy a barrier-method neighborhood condition.

## 5. Iterative refinement and end-to-end complexity

A direct dependence on the target accuracy $\zeta$ can enter through the conditioning of $AS^{-1}$. To avoid a linear $1/\zeta$ factor in $\kappa(AS^{-1})$, the algorithm embeds the low-precision quantum IPM inside an iterative-refinement outer loop.

At iterative-refinement step $k$, one solves a refining dual LP
\[
(D_{\mathrm{IR}}):\quad
\max_{\hat y,\hat s}\;\nabla^k b^T\hat y
\quad\text{s.t.}\quad
A^T\hat y+\hat s=\nabla^k s^{(k)},\;\hat s\ge 0,
\]
with barrier parameter
\[
\mu_{\mathrm{IR}}=(\nabla^k)^2\mu^0,
\]
to constant accuracy $\hat\zeta$. The resulting increment is scaled back by $1/\nabla^k$ and used to update the main iterate. By Lemmas 3.9–3.11, each iterative-refinement call requires $O(1)$ quantum-IPM steps, and only
\[
O\!\left(\frac{\log(1/\zeta)}{\log(1/\hat\zeta)}\right)
\]
outer iterative-refinement calls are needed to reach final tolerance $\zeta\ll \hat\zeta$ [2412.15977].

Combining the inexact-feasible DLBM analysis with iterative refinement yields the total QRAM-query complexity
\[
\widetilde{O}_{n,\|AS^{-1}\|_F,\kappa(AS^{-1}),\mu^0}\!\bigl(m\sqrt{n}\,\kappa^0\bigr),
\]
where
\[
\kappa^0=\kappa\!\bigl(A(S^0)^{-1}\bigr)^2
\]
is the condition number of the initial Newton system. The classical arithmetic cost is
\[
O\!\bigl(m\,n^{1.5}\log(n\mu^0/\zeta)\bigr).
\]

A particularly notable regime is specified explicitly: if the LP has $m$ constraints and $n$ variables with
\[
n\ge C\cdot m^2
\]
for some constant $C$, then
\[
m\sqrt{n}=O(n^{3/2}/\sqrt{m})=o(n),
\]
so the QRAM-query cost scales sublinearly in the dimension $n$ [2412.15977].

## 6. Related formulations, misconceptions, and practical assessment

The broader literature on quantum LP solvers includes several closely related formulations that differ primarily in how they preserve feasibility, how they manage condition-number growth, and whether they rely on Newton linearization at every step.

| Paper | Core formulation | Stated property |
|---|---|---|
| "An Inexact Feasible Interior Point Method for Linear Optimization with High Adaptability to Quantum Computers" [2307.14445] | Orthogonal Subspaces System (OSS) | Inexact solve preserves primal and dual feasibility; $O(\sqrt{n}L)$ iteration complexity |
| "An Inexact Feasible Quantum Interior Point Method for Linearly Constrained Quadratic Optimization" [2301.05357] | Feasible QIPM via orthogonal-subspace reduction | For LP specialization, $O(\sqrt{n}\log(1/\epsilon))$ iterations and per-iteration cost $O(\kappa(M))+O(n)$ |
| "Efficient Use of Quantum Linear System Algorithms in Interior Point Methods for Linear Optimization" [2205.01220] | Inexact infeasible QIPM | $O(n^2\log(1/\zeta))$ iterations; iterative refinement recovers exactness in polynomial time |
| "A preconditioned inexact infeasible quantum interior point method for linear optimization" [2412.11307] | Preconditioned reduced augmented system | Improves condition number from $O(1/\mu^2)$ to $O(1/\mu)$ |
| "Improvements to Quantum Interior Point Method for Linear Optimization" [2310.07574] | Preconditioned modified normal equations | Uses preconditioning with iterative refinement and $O(\log m+\log(1/\delta^k))$ qubit overhead |
| "A quantum central path algorithm for linear optimization" [2311.03977] | One-shot nonlinear central-path simulation | Avoids Newton iteration by simulating the full complementarity system |

One common misconception is that inexactness necessarily destroys feasibility. The feasible variants show otherwise. In the OSS-based approach, one sets $\Delta x=V\lambda$ with $V$ spanning $\mathrm{Null}(A)$ and $\Delta s=-A^T\Delta y$, so even an inexact solve preserves
\[
A(x+\alpha\widetilde{\Delta x})=b,\qquad
A^T(y+\alpha\widetilde{\Delta y})+(s+\alpha\widetilde{\Delta s})=c
\]
exactly for any $\alpha\in(0,1]$ [2307.14445]. The DLBM-based construction reaches the same objective from the dual side by imposing explicit bounds on $E_{\Delta s}^C$ and on the tomography error [2412.15977].

A second misconception is that asymptotic QLSA speedups are sufficient to establish a practical LP advantage. The practical lower-bound study "Practical lower bounds for hybrid quantum interior point methods in linear programming" evaluates a standard hybrid QIPM pipeline against the classical open-source solver HiGHS on a broad collection of LP instances and concludes that, across all instances and for any realistic quantum cycle duration, the quantum runtime lower bounds already exceed the classical runtimes [2604.24362]. This does not refute the asymptotic theory; rather, it isolates tomography, state loading, and end-to-end hybrid overhead as decisive obstacles.

A related lesson appears outside LP in "Approximate Quantum Linear Solvers for Hybrid CFD: End-to-End Analysis with a Chebyshev-LCU Approach." There, convergence of the outer nonlinear workflow is preserved for a non-exact quantum solver with only a moderate overhead in iteration count, provided that the high-frequency components of the linear system are resolved with sufficient accuracy; the approximate Cheb-LCU implementation reduces the required number of single-qubit rotations by over an order of magnitude relative to a QSVT-based solver while requiring only a modest increase in iteration count [2606.01067]. This suggests that end-to-end analysis, rather than isolated linear-solver complexity, is likely to remain central for assessing inexact quantum LP solvers as well.

In this landscape, the inexact quantum linear programming solver is best understood as a family of hybrid barrier and interior-point methods in which feasibility, neighborhood control, and precision management are co-designed with the quantum linear-system primitive. The DLBM-based solver of Wu, Sampourmahani, Mohammadisiahroudi, and Terlaky provides a particularly sharp realization of that program by coupling inexact-feasible analysis, QLSA-plus-tomography implementation, and iterative refinement into a single convergence theory with short-step iteration complexity [2412.15977].

Source: https://www.emergentmind.com/topics/inexact-quantum-linear-programming-solver